1. Introduction

At present, visible lasers have been applied to many fields, including laser display, medicine, optical storage, bio-photonics, undersea communication, marking and precision micro fabrications etc [1–4]. Based on the phase-matching (PM) technology of nonlinear optical (NLO) crystals, the frequency doubling (i.e. SHG) of near-infrared lasers has been an efficient and convenient approach to obtain such sources with high beam quality and reliability [5–7]. Among all of the PM configurations, the ones which propagate along the optical principal axes of the NLO crystal are very special, which are also known as angular non-critical phase-matching (A-NCPM) [8,9]. The A-NCPM possesses two distinctive advantages, i.e. the large angular acceptance bandwidth and the absence of the beam walk-off, so it is considered to be the optimal PM style. Therefore, in this condition long crystal can be utilized to increase the frequency conversion efficiency [10–15]. On the whole, the A-NCPM is rare and precious, since the fact that the refractive indexes of NLO crystals are usually unchangeable, which makes the A-NCPM wavelengths are almost fixed and limited. Such situation applies to most of the bulk NLO crystals, including the well-known KH₂PO₄ (KDP), KTiOPO₄ (KTP), BaB₂O₄ (BBO), and BiB₃O₆ (BIBO) [12]. By time now, the most popular A-NCPM NLO crystal is LiB₃O₅ (LBO), whose refractive indexes sensitively change with the temperature. On X-axis it can realize type-I SHG, the A-NCPM SHG wavelength range is 0.94~1.65 µm when the temperature varied in the range of −10~300 °C, and the effective NLO coefficient d_eff is 0.85 pm/V (d₃₂). On Z-axis LBO can realize type-II SHG, the NCPM wavelength range is 1.12~1.46 µm when the temperature varied in −50~50 °C, and the d_eff is 0.67 pm/V (d₃₁) [12]. Beside the temperature-dependence, the component dependence is another means to realize broad waveband A-NCPM. In 1999, Furuya et al. found that the ratio of lattice constants (a/b and a/c) for Gd_xY_1-xCOB crystals change linearly with the compositional parameter x. These results confirmed that Gd_xY_1-xCOB was a substitutional solid solution, and its refractive indices as well as A-NCPM wavelength could be continuously adjusted by varying the parameter x [10,16,17]. Subsequently, similar properties are also observed in other series of mixed RECOB crystals, such as Lu_xGd_1-xCOB and Sc_xGd_1-xCOB [8,18]. Comparing to the 0.94~1.65 µm A-NCPM SHG of LBO crystal, RECOB type crystals (where RE denotes a rare-earth element like Tm, Y, Gd, La, etc. or their “mixture”) can extend the PM wavelength toward short wave direction, which covers the work waveband of Ti:Sapphire lasers and near infrared laser diodes, i.e. 0.7~0.9 µm. Now, the A-NCPM frequency conversion has been one of the representative NLO applications of RECOB type crystals. Nevertheless, the related researches are still separated and scattered, the internal connection among different RECOB crystals has never been explored. Corresponding, the A-NCPM feature parameters, whether they are the experimental values in most cases or the theoretical calculated values in a few cases, have never been comprehensively investigated and compared.

In this paper, the A-NCPM SHG characteristics of RECOB (RE = Tm, Y, Gd, Sm, Nd and La) crystals are intensively studied. The single crystal growth, transmission spectra, orientation determination and refractive indices are shown in parts 2 to 5, as the necessary preparatory information. In parts 6 to 10, we discussed five A-NCPM SHG properties respectively, including the realizable styles, PM wavelength, angular acceptance, wavelength acceptance, and temperature acceptance. For the first time, the Sellmeier equations as well as the A-NCPM SHG properties of SmCOB, NdCOB crystals are reported. For the other four crystals, TmCOB, YCOB, GdCOB and LaCOB, the parameters presented in parts 2 to 7 are cited from the previous literatures which have been marked out in the text, and the experimental as well as the theoretical data in parts 8~10 are newly obtained in this work.

It should be noted that although NCPM technology include other manners like wavelength NCPM, temperature NCPM, we only concerned A-NCPM here. For simplicity, we use “NCPM” instead of “A-NCPM” in the followed parts.

2. Single crystal growth

Before Czochralski single crystal growth, the polycrystalline RECOB raw materials were prepared from high purity (4N) CaCO₃, Re₂O₃ (RE = Tm, Y, Gd, Sm, Nd and La) and H₃BO₃ powders. They were weighed referring to their nominal compositions. Considering the evaporation of B₂O₃ component during the solid-state reaction and crystal growth, an excess quantity of H₃BO₃ (1~4% in mass ratio) were added to the starting materials. After mixing the starting materials completely, the first calcination was carried out up to 950~1000 °C for 10~20 hours to decompose the H₃BO₃ and CaCO₃. Then the calcined powders were ground, remixed and pressed into tablets. Polycrystalline RECOB components were finally obtained by the second calcination at 1100~1200 °C for 15~25 hours. In order to make the components homogeneous enough for single crystal growth, the RECOB melts were kept 30~50 °C higher than their melting points (the melting points for different RECOB crystals are varied from 1390°C to1520 °C) for more than 12 hours. More crystal growth details can be found in previous reports [19–25]. We found that the TmCOB and LaCOB crystals were more difficult to grow comparing to other four RECOB (RE = Y, Gd, Sm and Nd) crystals. The LaCOB crystal is easy to crack, due to the approximate cleavage along the ($\overline{2}$01) planes, so its growth procedure needs to be carried out blandly. Different to the LaCOB crystals, the growth difficulty of the TmCOB crystals was induced by the narrow congruent region of TmCOB crystal. It was found that the deficient H₃BO₃ (excess amount < 3.0 wt%) or over added H₃BO₃ (excess amount > 4 wt%) would be harmful to high-quality TmCOB single crystals. Figure 1 shows the as-grown RECOB crystals that we have obtained. As the material basis of this paper, it can be seen that integrally they have large size and high optical quality. Similarly, the large size substitutional solid solutions with different RE components (hereafter we denote as “the mixed crystals”) can also be achieved by the Czochralski method, including Gd_xY_1-xCOB [16], Nd_xY_1-xCOB [26], Nd_xGd_1-xCOB [27], Nd_xLa_1-xCOB [28], and Sm_xY_1-xCOB [29].

 

Fig. 1 As-grown RECOB crystals (RE = Tm, Y, Gd, Sm, Nd and La).

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3. Transmission spectra

With a NIR-UV-VIS spectrophotometer (U-4001, Hitachi), the optical transmission spectra of RECOB type crystals were recorded at room temperature. All the samples were prepared along the <010> direction and were mechanical polished with a thickness of ~2 mm. The measuring scope was 190~1600 nm, and the scanning step was 1 nm. Figure 2 presents the optical transmission spectra of TmCOB, SmCOB and NdCOB crystals. Among the concerned RECOB (RE = Tm, Y, Gd, Sm, Nd and La) crystals, YCOB, GdCOB and LaCOB are colourless and transparent, which have no absorption peaks at 210~2500 nm, 320~2700 nm and 200~2400 nm, respectively [20,30,31]. The corresponding mixed crystals, Gd_xY_1-xCOB, La_xGd_1-xCOB and La_xY_1-xCOB are also colourless and transparent without absorption peak from ultra-violet (UV) to near infrared. The other three crystals, TmCOB, SmCOB and NdCOB, exhibit different colours, as shown in Fig. 1. They are relevant to different absorption peaks in visible waveband, as presented in Fig. 2. Such absorption characteristics are also inherited by the mixed crystals derived from them, like Sm_xY_1-xCOB and Nd_xGd_1-xCOB. The location and the width of the absorption peaks are almost the same, but the peak intensities will be obviously weaker, compared with SmCOB and NdCOB crystals. In Fig. 2, it is found that SmCOB exhibits less absorption peaks in visible region than TmCOB and NdCOB, with transmittance higher than 70% at 366~398 nm and 421~1054 nm. It means that SmCOB possesses wider spectral selecting space than TmCOB and NdCOB in NLO applications. The existence of absorption peaks in visible region for TmCOB, SmCOB, NdCOB and their derivative mixed crystals will affect their NLO applications more or less.

 

Fig. 2 Transmission spectra of several colored RECOB crystals.

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4. Crystal orientation

The determination of crystal orientation is the first step for investigating the NLO properties of RECOB crystals, which will directly affect the followed measurement of refractive indices and the calculation of NCPM characteristics. RECOB crystals are monoclinic biaxial materials which belong to the point group m. They possess three orthogonal optical principal axes X, Y and Z, among which the Y axis is reversed with the two-folded crystallographic axis b, and the other two axes X and Z locate in (010) plane which are respectively close to the crystallographic axes c and a with certain angles, as shown in Fig. 3. Consistent with the tradition, the X, Y, Z axes are assigned according to the relation for the principal refractive indices n_X < n_Y < n_Z. The RECOB crystals are prone to grow along b axis, and their ($\overline{2}$01), (010) and (101) crystallographic planes are usually exposed in this situation, which can help us for crystal orientation and sample preparation.

 

Fig. 3 Relationships between the crystallographic axes (a, b and c) and the optical principal axes (X, Y and Z) for RECOB crystals.

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The crystal orientation of TmCOB, YCOB, GdCOB and LaCOB had been reported previously, here we newly determined the values of SmCOB and NdCOB by using of the polarized microscopy (Axio Lab. A1, ZEISS), and all of the results were summarized in Table 1. Based on the orientation of RECOB crystals, the relationships between the crystallographic axes (a and c) and optical principal axes (Z and X) were found to be 23.5~27.8° for (a, Z) and 12.4~16.5° for (c, X), respectively. The angle β between the crystallographic axes a and c for all of the crystals unchanged basically, which was 101.2 ± 0.2°, as shown in Table 1. The light source used in this measurement is a He-Ne laser (632.8 nm). Because no dielectric frame rotation has ever been observed in such crystals [32,34], the data in Table 1 will be available for other wavelengths, either.

Table 1. Crystal orientation of RECOB crystals

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5. Refractive index

RECOB crystals are optical biaxial crystals, and each crystal needs to be processed into two right angle prisms whose ridges are along different optical principal axes, to permit the measurement of X, Y and Z polarization light, respectively. Utilizing a full-automatic, high-precision (2 × 10⁻⁶) refractive-index measurement instrument (Spectro Master @HR, Trioptics), the refractive indices of SmCOB, NdCOB, LaCOB crystals were measured, and the results were exhibited in Fig. 4. The refractive index data were obtained by fitting the dispersion curves according to the followed Sellmeier equations

 

Fig. 4 Refractive-index dispersion curves of several RECOB crystals.

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Table 2. Sellmeier equation coefficients of RECOB crystals

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6. NCPM SHG style

For biaxial NLO crystals, NCPM is the PM direction along the optical principal axes. RECOB crystals have three optical principal axes X, Y and Z, and each axis has two types of SHG PM styles, i.e. ssf (type-I) and sff (type-II), so in the ideal case they can have six NCPM SHG wavelengths at least. In view of the point group m and the Kleimann symmetry, the contracted matrix of nonlinear coefficients of RECOB crystals can be expressed as

Table 3. The effective nonlinear optical coefficients d_eff of RECOB type crystals along different optical principal axes. The values of the NLO coefficients came from Reference [37].

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From the viewpoint of NLO coefficient, the type-I NCPM on Y axis which has the largest d_eff of 0.60 pm/V should be the most favourable. It is close to the d_eff values of LBO crystal, i.e. 0.85 pm/V for type-I NCPM and 0.67 pm/V for type-II NCPM. Besides, the type-I NCPM on Y axis also has special material advantage, because RECOB crystals are easy to grow along the crystallography b axis which is collinear with the Y optical axis, so it is convenient to obtain long Y-cut samples to perform high efficiency NCPM SHG frequency conversions.

7. NCPM SHG wavelength

7.1 Calculation and experimental measurement

Using the Sellmeier equation parameters listed in Table 2, the NCPM SHG wavelengths of RECOB (Re = Tm, Y, Gd, Sm, Nd and La) crystals were calculated, and the results for practical NCPM styles, i.e. type-I and type-II on Y axis, and type-I on Z axis, were shown in Table 4. In order to examine these values, Y- and Z-cut RECOB samples were processed for experimental measurements. An OPO (Opolette HE 355 II) tunable laser was used as the fundamental laser source (410~2400 nm). The output light was linearly polarized along the vertical direction. The laser pulse width was about 5 ns and the repetition rate was 20 Hz. The average output power depended on the work wavelength and driving current, which varied in the range of 0~12 mW. For Y-cut RECOB crystal samples, the polarization of the Y-transmitted fundamental wave was along the Z-axis for type-I NCPM SHG (i.e. slow wave polarization), but along the angle bisector of Z-axis and X-axis for type-II NCPM SHG (i.e. slow wave polarization plus fast wave polarization). For Z-cut RECOB crystal samples, the polarization of the Z-transmitted fundamental wave was along the Y-axis for type-I NCPM SHG (i.e. slow wave polarization). When the transmitting direction and polarization state of the fundamental wave was correctly arranged, the accurate NCPM SHG wavelength could be determined by tuning the output wavelength of the OPO laser to find which one yielded the maximum SHG output signal. The NCPM SHG wavelengths for different RECOB crystals were measured and listed in Table 4.

Table 4. Calculated and measured NCPM SHG wavelength of RECOB crystals

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In this study, the type-II NCPM SHG wavelengths on Y-axis for SmCOB and NdCOB crystals didn’t obtained, which might attribute to the low output power at the infrared waveband of the OPO laser, and the strong absorption of fundamental wave around 1.4 µm for SmCOB, as well as the strong absorption of SHG wave around 0.8 µm for NdCOB, as shown in Fig. 2. By comparing the calculated results with the measured results in Table 4, it can be concluded that the Sellmeier equations of SmCOB, NdCOB, and LaCOB crystal which we reported in this paper are quite accurate. The discrepancies between the calculated data and the experimental values are less than 10 nm, and in some conditions they are only 2 nm. At the same time, for the other three crystals TmCOB, YCOB and GdCOB the wavelength discrepancies between calculations and experiments are also satisfactory basically, which reflects the accuracy of the corresponding Sellmeier equations. The largest difference comes from the type-II NCPM on Y-axis of GdCOB crystal, which is found to be 50 nm. Generally, for type-I NCPM SHG wavelengths the experimental values are found to agree well with the calculated ones, while for the type-II NCPM SHG wavelengths the errors are more obvious which can reach several tens of nanometers. The main reason for this phenomenon is that for type-II PM there are three different refractive indexes (slow fundamental wave, fast fundamental wave and fast SHG wave) which participate the calculation of PM wavelength, while for type-I PM only two different refractive indexes participate the calculation (slow fundamental wave and fast SHG wave), so the precise prediction of type-II PM property proposes higher requirement for the accuracy of Sellmeier equations. In addition, the environmental temperature discrepancy between the refractive index measurements and the NCPM SHG experiments will also cause the wavelength deviations between calculation and measurement.

For multiple components RECOB, i.e. mixed RECOB crystals, the NCPM wavelength can be adjusted continuously by varying the component parameter x. Taking Sm_xY_1-xCOB series crystals for example, Table 4 indicates that theoretically the type-I and type-II NCPM SHG wavelengths adjusting ranges on Y-axis are 723~903 nm and 1023~1390 nm, respectively, and the type-I NCPM wavelength adjusting range on Z-axis is 827~1123 nm. It means the spectral range from 0.72 to 1.39 µm can be absolutely covered by Sm_xY_1-xCOB crystals.

7.2 Influence of rare earth ions RE³⁺

For single component and mixed RECOB crystals, the difference of rare earth ions RE³⁺ leads to the variation of optical properties, including the transmission spectrum, the orientation of the optical principle axis, and the refractive index. Accordingly, the refractive index dispersion decides the NCPM SHG wavelength directly. So it is necessary to explore the relationships among the type of RE³⁺ ions, the refractive index, and the NCPM SHG wavelength. A theoretical analysis indicates that the NCPM SHG wavelength takes on the opposite change trend with the birefringence of the fundamental wave, which is proved in the first part of Appendix (12.1) in details. In Figs. 5a, 5b and 5c, we plotted the variation of birefringence and NCPM SHG wavelength with rare earth radius for the three available NCPM SHG styles, respectively. Referencing Table 4, we chose 800, 1300 and 1064 nm as the representative fundamental wavelengths for type-I, type-II NCPM SHG on Y-axis and type-I NCPM SHG on Z-axis, respectively. The corresponding birefringence of different RECOB crystals were plotted in Figs. 5a-5c. Because we have not experimentally found the type-II NCPM SHG wavelengths on Y-axes of SmCOB and NdCOB crystals, the calculated data of 1390, 1578 nm for them were used in Fig. 5b. Other NCPM SHG wavelengths in Figs. 5a~5c were adopted with the experimental data in Table 4. For RECOB crystals, the ionic radius of Tm³⁺, Y³⁺, Gd³⁺, Sm³⁺, Nd³⁺, La³⁺ and Ca²⁺ positive ions are 0.880, 0.890, 0.938, 0.958, 0.983, 1.032 and 0.995 Å, respectively. When the radius of the rare earth ion RE³⁺ increases, the average bond length perpendicular to the plane of RE-O bond will rapidly increases, either, which weakens the electronegativity of RE element, decreases the crystal melting temperature, increases the crystal lattice stress, and elevates the risk of cracking [21]. The crystal growths have manifested that LaCOB is easier to crack than other five RECOB (Tm, Y, Gd, Sm and Nd) crystals.

 

Fig. 5 Variation of birefringence and NCPM SHG wavelength with rare earth ion radius in RECOB crystals.

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Figures 5a, 5b and 5c exhibit the same variation discipline between the ionic radius of RE³⁺ and the birefringence, i.e. the closer ion radius of RE³⁺ and Ca²⁺, the smaller birefringence. At the same time, the experimental results are accordant with the theoretical derivations of the first part of the appendix (12.1), i.e. the small crystal birefringence corresponds to the long NCPM SHG wavelength. So there is the closer ion radius of RE³⁺ and Ca²⁺, the longer NCPM SHG wavelength. In Figs. 5a-5c, NdCOB possesses the smallest birefringence, correspondingly its NCPM SHG wavelengths are the longest, which are 927, 1578 and 1205 nm, respectively. TmCOB possesses the largest birefringence, so its NCPM SHG wavelengths are the shortest, which are 716, 1011 and 815 nm, respectively. As the optimum NCPM style of RECOB type crystals which has the largest d_eff, the type-I NCPM SHG on Y-axis presents a tuning amplitude larger than 200 nm, as shown in Fig. 5a. With the increase of ionic radius of RE³⁺, the birefringence decreases from 0.044 to 0.029 and then increases to 0.035, correspondingly, the NCPM SHG wavelength increases from 716 nm to 927 nm and then decreases to 808 nm (Fig. 5a). The type-II NCPM SHG wavelength on Y-axis changes from 1011 nm to 1578 nm and then decreases to 1178 nm with the increase of ion radius of RE³⁺ (Fig. 5b). The type-I NCPM SHG wavelength on Z-axis changes from 815 nm to 1205 nm and then decreases to 1001 nm (Fig. 5c).

In short, in this part we revealed the NCPM SHG wavelength variation discipline of RECOB crystals for the first time, i.e. it depends on the birefringence. The more essential reason is the radius difference between the rare earth ion RE³⁺ and the lattice positive ion Ca²⁺, not the absolute radius of RE³⁺. The basic principle for crystal design can be summarized as: the closer ion radius between RE³⁺ and Ca²⁺, the longer NCPM SHG wavelength.

8. Angular acceptance bandwidth

Besides of d_eff and PM wavelength, various acceptance bandwidths are also known as the basic attributes of NLO crystal. According to the different influence parameters, the acceptance bandwidths are mainly divided into three categories, i.e. angular, wavelength, and temperature, which relate to the spatial dispersion, frequency dispersion, and temperature dispersion of refractive indexes, respectively. In the followed three parts, we will discuss them one by one, aiming at the NCPM SHG configurations in RECOB crystals. For the angular NCPM SHG concerned in this paper, extremely large angular acceptance bandwidth is one of the most important advantages. Unfortunately, by time now no accurate mathematic expressions of this parameter are reported for NCPM SHG situations, to the best of our knowledge. The main reason is they are related to the second derivative of the wave vector mismatching, which are complicated and difficult to deduce. In the second part of the appendix (12.2), we give a complete theoretical deduction about these expressions for biaxial NLO crystal, and the results are summarized in the Table 8. As we have known it is the first time that the angular acceptance bandwidths of NCPM SHG styles are obtained directly in theory. In the past, people got these parameters only by indirectly experimental measurement. If the condition of n_x = n_y is substituted in these expressions, the results in Table 8 are also available for uniaxial NLO crystal. Therefore, the deduction and the results of the second part of the appendix (12.2) are meaningful for all of the anisotropic NLO crystals.

Using the expressions in Table 8, we calculated the internal angular acceptance bandwidths for the NCPM SHG of different RECOB crystals, based on the Sellmeier equation coefficients listed in Table 2. Then referencing the sine law of refraction and the refractive indexes of various crystals at their NCPM SHG wavelengths, we obtained the external angular acceptance bandwidths in theory, which were shown in Table 5. As discussed in the part 6, only three NCPM SHG styles are realizable in these crystals, including type-I, type-II on Y axis, and type-I on Z axis. The grey blocks represent unavailable NCPM SHG style (type-II on Z axis), and unavailable angular acceptance bandwidth (Δϕ⋅l^1/2 on Z axis). As introduced in the second part of the appendix (12.2), there is only angular acceptance bandwidth of Δθ⋅l^1/2, and no Δϕ⋅l^1/2 for the NCPM on Z axis. Beside of the theoretical calculation, we also actually measured the angular acceptance bandwidths for type-I NCPM SHG on Y axis, including Δθ⋅l^1/2 and Δϕ⋅l^1/2, for five RECOB crystals. The external angular tuning curves of the SHG signals are shown in Fig. 6, and the corresponding angular acceptance bandwidths are listed in Table 5. Because TmCOB has severe absorption for its type-I NCPM SHG output on Y axis (358 nm), we cannot accurately measure the variation of SHG signal in this crystal, so the corresponding experimental data of TmCOB crystal are unavailable in Fig. 6 and Table 5. In addition, we measured the angular acceptance bandwidths of type-I NCPM SHG on Z axis, for LaCOB crystal. The results are listed in Table 5 as well. It can be seen that all of the experimental data are in good agreement with the calculated data, with discrepancies less than 5 mrad·cm^1/2. It indicates that the theoretical calculation formulas obtained in the second part of the appendix (12.2) are correct, and the refractive index dispersion equations shown in part 5 are precise.

Table 5. External angular acceptance bandwidth of NCPM SHG in RECOB crystals

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Fig. 6 External angular tuning curves of the SHG signals in different RECOB crystals, for the type-I NCPM on Y axis.

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From Table 5, we can find the followed disciplines about the external angular acceptance bandwidth of RECOB crystals:

9. Wavelength acceptance bandwidth

For various angular NCPM SHG styles, the mathematical expressions of the wavelength acceptance bandwidth Δλ⋅l are discussed in the third part of the appendix (12.3) and the results are summarized in the Table 9. Combining with the Sellmeier equation coefficients listed in Table 2, we calculated Δλ⋅l for the six RECOB (RE = Tm, Y, Gd, Sm, Nd and La) crystals, as shown in Table 6. It can be seen that the Δλ⋅l of type-II NCPM SHG is larger than the values of type-I NCPM. At the same time for type-I NCPM SHG the Δλ⋅l on the Z axis is larger than that on the Y axis. Such discipline can be intuitively explained by the corresponding formula in Table 9, considering the property of B_Z > B_Y, D_Z > D_Y (Table 2). Among all of the six RECOB crystals, the largest Δλ⋅l comes from NdCOB, and the smallest comes from TmCOB, which can attribute to the radius difference between the rare earth ion RE³⁺ and the lattice positive ion Ca²⁺. As shown in Fig. 5, the Nd³⁺ ion radius is the closest to the Ca²⁺ ion radius, correspondingly the birefringence of NdCOB is the smallest, so the Δλ⋅l is the largest. On the contrary, the Tm³⁺ ion radius is the farthest to the Ca²⁺ ion radius, so the birefringence of TmCOB is the largest and the Δλ⋅l is the smallest. For Gd³⁺ and La³⁺, their ion radiuses have similar discrepancy to the Ca²⁺ ion radius, so GdCOB and LaCOB exhibited similar birefringence and wavelength acceptance bandwidth. In summary, the radius difference between Re³⁺ and Ca²⁺ is still the decisive factor for the magnitude of wavelength acceptance bandwidth, i.e. the small radius difference corresponds to the small birefringence and large Δλ⋅l.

Table 6. Wavelength acceptance bandwidth of NCPM SHG in different RECOB crystals

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By fixing RECOB crystals at angular NCPM SHG conditions, we measured wavelength tuning curves of the SHG signals, with an OPO laser as the fundamental light sources. Four representative experimental curves were shown in Fig. 7. All of the measured results of the wavelength acceptance bandwidths are presented in Table 6. Limited by the experimental conditions or the crystal’s characteristics, some of the measured values were not available, and the reasons were individually explained as the annotations of the Table 6. It can be seen that the calculated values are in good agreements with those measured from the experiments, which proves the reliability of the theoretical deduced formula in Table 9.

 

Fig. 7 Wavelength tuning curves of the SHG signals in different RECOB crystals, for the type-I SHG on Y axis.

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10. Temperature acceptance bandwidth

A discussion on the temperature acceptance bandwidth for the NCPM SHG of biaxial crystal is given in the fourth part of the appendix (12.4), and the theoretically deduced results are presented in the Table 10. Based on the thermo-optic dispersion formula of YCOB and GdCOB crystals [43,44], the temperature acceptance bandwidth for these two crystals are calculated out, as shown in the Table 7. It can be seen that for Type-I NCPM SHG our calculation values are basically in agreement with the experimental data reported in the references [43,44]. Although there are no measured data about the Type-II NCPM SHG of YCOB and GdCOB crystals, the existing experimental results of Gd_0.132Y_0.868COB crystal (Y-axis, ω = 1053 nm, ΔT⋅l = 43.5 °C·cm) [45] and La_0.1204Y_0.8796COB crystal (Y-axis, ω = 1064 nm, ΔT⋅l = 38.4 °C·cm) [46] are close to the calculating values in the Table 7, which can prove the validity of the theoretical deductions. As mentioned in the references [44], the data discrepancy between calculation and experiment may come from three aspects: (1) The thermal rotations of optical indicatrix axes in RECOB type crystals, which have been ignored during the formulae deduction procedures of the Table 10. (2) The crystal-to-crystal variation of the thermo-optic constants. (3) The Gd or La molar concentration gradient in the mixed crystals. Umemura et al ever measured the 90° phase-matching temperature for type-I third-harmonic-generation at 355 nm by using three y-cut Gd0.32Y0.78OB crystals. Although their index and thermo-optic formulas for YCOB and GdCOB give the 90° phase-matching temperature of Tpm ∇ 109 °C, the experimental values were Tpm ∇ 85 ± 1 °C, 93 ± 1 °C, and 102 ± 1 °C, respectively [44]. This example proved that the temperature characteristics of RECOB crystals do exist large individual difference.

Table 7. Temperature acceptance bandwidth of NCPM SHG in YCOB and GdCOB crystals

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On the whole, the temperature acceptance bandwidth of RECOB crystals are much larger than that of LBO crystal, such property can reduce the requirement of the temperature controlling and is favourable for high stability NLO frequency conversions. As early as 2002, we have confirmed this property by SHG experiments of 1064 nm laser. For this frequency conversion of YCOB and GdCOB crystals, the PM angle which is closest to the optical principle axes is (90°, 75.4°), corresponding to the Type-II PM of YCOB crystal. So it is the most promising angle to achieve NCPM by changing the crystal temperature. Using an extra cavity frequency doubling experiment equipment and a (90°, 90°)-cut (i.e. Y-cut) YCOB sample, we monitored the changing of this PM angle with the crystal temperature. Although this PM angle was indeed gradually close to the Y axis with the increasing of temperature, but the speed was very slow. When the crystal temperature was elevated to 150°C from a room temperature of 20 °C, this PM direction only changed about 6°, and the angular distance from Y-axis was still 9° or so [47]. At the same temperature variation range, the PM angle of LBO crystal will change 12°, and the one of MgO:LiNbO₃ crystal will change more than 14°, which have realized A-NCPM SHG [48]. On the PM direction (31°, 180°) of YCOB crystal, the temperature acceptance bandwidth is found to be even larger than 150°C, reported by A. Kausas et al very recently [49].

In the future, as long as the thermo-optic dispersion formula of other RECOB (RE = Sm, Nd, La and Tm) crystals are supplied, according to the Table 10 their temperature acceptance bandwidths can be calculated out and supplemented to the Table 7.

11. Conclusion

As a kind of relatively new NLO material, the excellent comprehensive performance makes RECOB type crystals have many practical applications. The members of RECOB type crystals are numerous. RE³⁺ ions can be one or more kinds of trivalent rare-earth cations, in addition, Ca²⁺ ions can be partially replaced by two valent cations [50,51]. The crystal refractive indexes can be adjusted in a large scope by changing the components, correspondingly the PM properties can be changed dramatically. This advantage of RECOB type crystals brings the important NLO application, i.e. the broad waveband tunable NCPM.

In this paper, we intensively investigated the NCPM SHG properties of six RECOB (RE = Tm, Y, Gd, Sm, Nd and La) crystals. Based on the experimental results and the theoretical analysis, we discovered an important discipline that the closer ion radius between RE³⁺ and Ca²⁺, the smaller birefringence, and the longer NCPM SHG wavelength, the better angular and wavelength acceptance bandwidths. The analytical formula of the characteristic parameters for NCPM SHG were obtained by theoretical derivations. Referencing the refractive index attributes, the calculation results agreed well with the experimental data. The present research illustrates that RECOB type crystals are potential NCPM SHG materials for various near infrared wavelengths. At the same time, the discipline that we have discovered between the crystal component and the NCPM characteristics will contribute to the design of novel, excellent NCPM materials in the future.

12. Appendix

12.1 A discussion about the relationship between birefringence and NCPM SHG wavelength

(1) Type-I NCPM SHG on Y-axis

For this PM style, the refractive indexes of the NLO crystal should satisfy the condition

(3)$$n_Z(\omega )=n_X(2\omega )$$

where n_z(ω), n_x(2ω) are the refractive index on Z-axis for the fundamental wave, on X-axis for the SHG wave, respectively. Combining with the Eq. (A.1.1) in the main text, we can obtain

(4)$$n_Z{(\omega )}^2-n_X{(\omega )}^2=n_X{(2\omega )}^2-n_X{(\omega )}^2$$

(5)$$=\frac{B_X}{{(\frac{\lambda }{2})}^2-C_X}-D_X{\left(\frac{\lambda }{2}\right)}^2-(\frac{B_X}{{\lambda }^2-C_X}-D_X{\lambda }^2)$$

(6)$$=\frac{3B_X{\lambda }^2}{({\lambda }^2-4C_X)({\lambda }^2-C_X)}+\frac{3}{4}{D}_X{\lambda }^2$$

where λ is the wavelength of the fundamental wave when the type-I NCPM SHG is realized. From Table 3, it can be known that for various RECOB type crystals there are B_x = 0.022 ( ± 0.002), C_x = 0.016 ( ± 0.003), D_x = 0.006 ( ± 0.003). From Tables 5 and 6, it can be known that λ is 0.8 ( ± 0.1) μm. Taking these values into the right side of the Eq. (6), the first term and the second term change to 0.1836λ² and 0.0045λ², respectively. So the first term is about 40 times of the second term, the second term can be omitted approximately. Since λ² is also about 40 times of C_x, we can further simplify the coefficient (λ²-C_x) of the first term to be λ². Considering above approximations, the Eq. (6) can be rewritten as

(7)$$\Delta n(\omega )=n_Z(\omega )-n_X(\omega )\approx \frac{3B_X}{{\lambda }^2-4C_X}\times \frac{1}{n_Z(\omega )+n_X(\omega )}$$

From the Sellmeier equation coefficients of Table 3, we can calculate n_Z(ω)-n_X(ω) and n_Z(ω) + n_X(ω) at 0.8 μm of these crystals, which are 0.365 ± 0.075, 3.385 ± 0.015, respectively. Comparing with the ± 20.5% variation of n_Z(ω)-n_X(ω), the ± 0.44% variation of n_Z(ω) + n_X(ω) is much smaller, which can be regarded as a constant term that is independent with ω or λ. Under this approximation, there is

(8)$$\Delta n(\omega )=n_Z(\omega )-n_X(\omega )\propto \frac{3B_X}{{\lambda }^2-4C_X}$$

It indicates that the birefringence Δn and NCPM SHG wavelength λ have approximate inverse relationship, i.e. small birefringence will present long NCPM SHG wavelength.

(2) Type-II NCPM SHG on Y-axis

For this PM style, the refractive indexes of the NLO crystal should satisfy the condition

(9)$$\frac{n_Z(\omega )+n_X(\omega )}{2}=n_X(2\omega )$$

where n_z(ω), n_x(ω) are the Z-polarization, X-polarization refractive indexes of the fundamental wave, respectively. From the Eq. (9), we can obtain

(10)$${\left(\frac{n_Z(\omega )+n_X(\omega )}{2}\right)}^2-n_X{(\omega )}^2=n_X{(2\omega )}^2-n_X{(\omega )}^2$$

Referencing the Eqs. (4) - (7), there is

(11)$${\left(\frac{n_Z(\omega )+n_X(\omega )}{2}\right)}^2-n_X{(\omega )}^2\approx \frac{3B_X}{{\lambda }^2-4C_X}$$

i.e.

(12)$$\Delta n(\omega )=n_Z(\omega )-n_X(\omega )\approx \frac{3B_X}{{\lambda }^2-4C_X}\times \frac{4}{n_Z(\omega )+3n_X(\omega )}$$

Based on the similar analysis as above part, it can be known that the variation of n_Z(ω) + 3n_X(ω) is relatively small, so it can be recognized as a constant term. Thus, there is

(13)$$\Delta n(\omega )=n_Z(\omega )-n_X(\omega )\propto \frac{3B_X}{{\lambda }^2-4C_X}$$

It indicates that in this condition the birefringence Δn and NCPM SHG wavelength λ also have approximate inverse relationship.

(3) Type-I NCPM SHG on Z-axis

For this PM style, the refractive indexes of the NLO crystal should satisfy the condition

(14)$$n_Y(\omega )=n_X(2\omega )$$

Experiencing a similar discussion procedure with that of type-I NCPM SHG on Y-axis, we obtain

(15)$$\Delta n(\omega )=n_Y(\omega )-n_X(\omega )\propto \frac{3B_X}{{\lambda }^2-4C_X}$$

So for this PM style, the birefringence Δn and NCPM SHG wavelength λ still have inverse relationship.

12.2 Angular acceptance bandwidth of NCPM SHG

When the actual situations deviate from the ideal PM conditions, a mismatch of wave vector Δk will appear, and the SHG conversion efficiencyη can be expressed as

(16)$$\eta ={\eta }_0{\left[\frac{\sin \left(\frac{\Delta k}{2}l\right)}{\frac{\Delta k}{2}l}\right]}^2$$

where η₀ is the maximum SHG conversion efficiency under the ideal PM conditions, i.e. Δk = 0, and l is the transmission length of the NLO medium. When Δk = ± π/l, η will decreases to 40% or so of η₀, and the corresponding parameter value in Δk is defined as the acceptance of this parameter. It is 13% greater than the full width at half maximum of the sinc²(Δkl/2) function in Eq. (16), which corresponds to a 50% decreasing of η₀. Here we firstly consider the situation about the spatial angle θ. Around the PM angle (θ_m, ϕ_m), Δk can be expanded to the followed Taylor series about the θ angle

(17)$$\Delta k=\Delta k|{}_{\left(\theta ={\theta }_m,\varphi ={\varphi }_m\right)}+\frac{\partial \Delta k}{\partial \theta }|{}_{\left(\theta ={\theta }_m,\varphi ={\varphi }_m\right)}\Delta \theta +\frac{1}{2}\frac{{\partial }^2\Delta k}{\partial {\theta }^2}|{}_{\left(\theta ={\theta }_m,\varphi ={\varphi }_m\right)}\Delta {\theta }^2\cdots$$

So the angler acceptance in θ direction, i.e. Δθ, can be theoretically deduced from the Eq. (17) and the definition of Δk = ± π/l, if $\partial \Delta k/\partial \theta$ and ${\partial }^2\Delta k/\partial {\theta }^2$ is known. The followed part is a detailed deduction of these two parameters.

For three wave frequency mixing, based on the expression of k = 2πn/λ, there is

(18)$$\Delta k=k_3-k_2-k_1=\frac{2\pi }{{\lambda }_3}{n}_3-\frac{2\pi }{{\lambda }_2}{n}_2-\frac{2\pi }{{\lambda }_1}{n}_1$$

So

(19)$$\frac{\partial \Delta k}{\partial \theta }=\frac{2\pi }{{\lambda }_3}\frac{\partial n({\omega }_3)}{\partial \theta }-\frac{2\pi }{{\lambda }_2}\frac{\partial n({\omega }_2)}{\partial \theta }-\frac{2\pi }{{\lambda }_1}\frac{\partial n({\omega }_1)}{\partial \theta }$$

(20)$$\frac{{\partial }^2\Delta k}{\partial {\theta }^2}=\frac{2\pi }{{\lambda }_3}\frac{{\partial }^{2}n({\omega }_3)}{\partial {\theta }^2}-\frac{2\pi }{{\lambda }_2}\frac{{\partial }^{2}n({\omega }_2)}{\partial {\theta }^2}-\frac{2\pi }{{\lambda }_1}\frac{\partial n^2({\omega }_1)}{\partial {\theta }^2}$$

It is known that the polarized refractive index of anisotropic crystal on the light transmission direction (θ, ϕ) has the followed form

(21)$$n({\omega }_i)={\left(\frac{2}{-B_i\pm \sqrt{B_i^2-4C_i}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}i=1,2,3$$

For type-I PM the sign in above equation is “−” for i = 1, 2 which represents the slow fundamental wave, and the sign is “+” for i = 3 which represents the fast mixing frequency wave. Correspondingly, for type-II PM the sign is “−” for i = 1 which represents the slow fundamental wave, and the sign is “+” for i = 2, 3 which represents the fast fundamental wave and the fast mixing frequency wave, respectively. In the Eq. (21), there is

(22)$$\begin{array}{l}{B}_i=-{\sin }^2\theta {\cos }^2\varphi \left[n_y^{-2}({\omega }_i)+n_z^{-2}({\omega }_i)\right]\\ -{\sin }^2\theta {\sin }^2\varphi \left[n_x^{-2}({\omega }_i)+n_z^{-2}({\omega }_i)\right]-{\cos }^2\theta \left[n_x^{-2}({\omega }_i)+n_z^{-2}({\omega }_i)\right]\end{array}$$

(23)$$\begin{array}{l}{C}_i={\sin }^2\theta {\cos }^2\varphi n_y^{-2}({\omega }_i)n_z^{-2}({\omega }_i)+{\sin }^2\theta {\sin }^2\varphi n_x^{-2}({\omega }_i)n_z^{-2}({\omega }_i)\\ +{\cos }^2\theta n_x^{-2}({\omega }_i)n_z^{-2}({\omega }_i)\end{array}$$

From the Eq. (21), it can be obtained that

(24)$$\frac{\partial n_i}{\partial \theta }=-\frac{\sqrt{2}}{2}{\left(-B_i\pm \sqrt{B_i^2-4C_i}\right)}^{-\frac{3}{2}}\left(-\frac{\partial B_i}{\partial \theta }\pm \frac{B_i\frac{\partial B_i}{\partial \theta }-2\frac{\partial C_i}{\partial \theta }}{\sqrt{B_i^2-4C_i}}\right)$$

(25)$$\begin{array}{l}\frac{{\partial }^2{n}_i}{\partial {\theta }^2}=-\frac{\sqrt{2}}{2}\left\{{\left(-B_i\pm \sqrt{B_i^2-4C_i}\right)}^{-\frac{5}{2}}\left(-\frac{3}{2}\right){\left(-\frac{\partial B_i}{\partial \theta }\pm \frac{B_i\frac{\partial B_i}{\partial \theta }-2\frac{\partial C_i}{\partial \theta }}{\sqrt{B_i^2-4C_i}}\right)}^2\right\}\\ -\frac{\sqrt{2}}{2}{\left(-B_i\pm \sqrt{B_i^2-4C_i}\right)}^{-\frac{3}{2}}\left\{-\frac{{\partial }^2{B}_i}{\partial {\theta }^2}\pm \left[\left({\left(\frac{\partial B_i}{\partial \theta }\right)}^2+B_i\frac{{\partial }^2{B}_i}{\partial {\theta }^2}-2\frac{{\partial }^2{C}_i}{\partial {\theta }^2}\right)\frac{1}{\sqrt{B_i^2-4C_i}}-{\left(B_i\frac{\partial B_i}{\partial \theta }-2\frac{\partial C_i}{\partial \theta }\right)}^2{\left(B_i^2-4C_i\right)}^{-\frac{3}{2}}\right]\right\}\end{array}$$

where

(26)$$\frac{\partial B_i}{\partial \theta }=\left[n_x^{-2}({\omega }_i)+n_y^{-2}({\omega }_i)-n_z^{-2}({\omega }_i)-{\cos }^2\varphi n_y^{-2}({\omega }_i)-{\sin }^2\varphi n_x^{-2}({\omega }_i)\right]\sin 2\theta$$

(27)$$\frac{{\partial }^2{B}_i}{\partial {\theta }^2}=\left[n_x^{-2}({\omega }_i)+n_y^{-2}({\omega }_i)-n_z^{-2}({\omega }_i)-{\cos }^2\varphi n_y^{-2}({\omega }_i)-{\sin }^2\varphi n_x^{-2}({\omega }_i)\right]2\cos 2\theta$$

(28)$$\frac{\partial C_i}{\partial \theta }=\left[n_y^{-2}({\omega }_i)n_z^{-2}({\omega }_i){\cos }^2\varphi +n_x^{-2}({\omega }_i)n_z^{-2}({\omega }_i){\sin }^2\varphi -n_x^{-2}({\omega }_i)n_y^{-2}({\omega }_i)\right]\sin 2\theta$$

(29)$$\frac{{\partial }^2{C}_i}{\partial {\theta }^2}=\left[n_y^{-2}({\omega }_i)n_z^{-2}({\omega }_i){\cos }^2\varphi +n_x^{-2}({\omega }_i)n_z^{-2}({\omega }_i){\sin }^2\varphi -n_x^{-2}({\omega }_i)n_y^{-2}({\omega }_i)\right]2\cos 2\theta$$

At NCPM conditions there is θ = θ_m = 0° or 90°, so $\partial B_i/\partial \theta$ and $\partial C_i/\partial \theta$ will be zero because the Eqs. (26) and (28) contain the term of “sin2θ”. So $\partial n_i/\partial \theta$ and $\partial \Delta k/\partial \theta$ will be zero from the Eqs. (24) and (19), respectively. Simultaneously considering $\Delta k|{}_{(\theta ={\theta }_m,\phi ={\phi }_m)}=0$ at angular PM conditions, the right side of the Eq. (17) will only left the third term, and the angler acceptance Δθ can be obtained from the followed formula

(30)$$\Delta k=\frac{1}{2}\frac{{\partial }^2\Delta k}{\partial {\theta }^2}|{}_{(\theta ={\theta }_m,\phi ={\phi }_m)}\Delta {\theta }^2=\pm \frac{\pi }{l}$$

i.e. the angler acceptance bandwidth in θ direction for NCPM can be expressed as

(31)$$\Delta \theta \cdot l^{\frac{1}{2}}={\left[4\pi /\frac{{\partial }^2\Delta k}{\partial {\theta }^2}|{}_{(\theta ={\theta }_m,\phi ={\phi }_m)}\right]}^{\frac{1}{2}}$$

where ${\partial }^2\Delta k/\partial {\theta }^2$ can be obtained from the Eqs. (20), (25), (27) and (29). For example, to the type-I NCPM on Y-axis of biaxial crystal, i.e. (θ_m = 90°, ϕ_m = 90°), there is

(32)$$\begin{array}{l}\frac{{\partial }^2{n}_i}{\partial {\theta }^2}=-\sqrt{2}{\left[n_x^{-2}({\omega }_i)+n_z^{-2}({\omega }_i)\pm \left(n_x^{-2}({\omega }_i)-n_z^{-2}({\omega }_i)\right)\right]}^{-\frac{3}{2}}\\ \left[n_y^{-2}({\omega }_i)-n_z^{-2}({\omega }_i)\pm \left(n_z^{-2}({\omega }_i)-n_y^{-2}({\omega }_i)\right)\right]\end{array}$$

At SHG condition (ω₁ = ω₂), according to the sign selection rules mentioned above, we can get

(33)$$\frac{{\partial }^2{n}_1}{\partial {\theta }^2}=\frac{{\partial }^2{n}_2}{\partial {\theta }^2}=-n_z^3({\omega }_1)\left(n_y^{-2}({\omega }_1)-n_z^{-2}({\omega }_1)\right)i=1,2$$

(34)$$\frac{{\partial }^2{n}_3}{\partial {\theta }^2}=0i=3$$

So the angular acceptance bandwidth in θ direction for such NCPM SHG configuration can be deduced from the Eqs. (20) and (31), which is

(35)$$\Delta \theta \cdot l^{\frac{1}{2}}=10\sqrt{\frac{{\lambda }_1}{2n_z^3({\omega }_1)(n_y^{-2}({\omega }_1)-n_z^{-2}({\omega }_1))}}\begin{array}{cc}& \end{array}(mrad\cdot c{m}^{\frac{1}{2}})$$

Similarly, the angular acceptance bandwidth in θ direction for other five kinds of NCPM SHG configurations can be deduced out, which are listed in the Table 8.

Table 8. Angular acceptance bandwidth of NCPM SHG in biaxial NLO crystal

View Table | View all tables in this article

Based on the similar deduction procedures, we obtain the angular acceptance bandwidth in ϕ direction for all of the six kinds of NCPM SHG configurations, and list them in Table 8 as well. For the NCPM on Z axis (θ = 0), the change of spatial direction is only related to the parameter θ, so the angular acceptance bandwidths on the two principal planes, XZ and YZ, are both recorded as Δθ⋅l^1/2, which are distinguished by PM angles, i.e. (0°, 0°), (0°, 90°), respectively. It can be seen that on the same principal axis the angular acceptance bandwidth of type-II NCPM is usually larger than that of type-I NCPM, which exhibits a $\sqrt{2}$ times increase.

12.3 Wavelength acceptance bandwidth of NCPM SHG

For three wave frequency mixing, the three wavelengths satisfy the followed formula

(36)$$\frac{1}{{\lambda }_1}+\frac{1}{{\lambda }_2}=\frac{1}{{\lambda }_3}$$

If the fundamental wavelengths have the offsets Δλ₁ and Δλ₂, the above formula will be

(37)$$\frac{1}{{\lambda }_1+\Delta {\lambda }_1}+\frac{1}{{\lambda }_2+\Delta {\lambda }_2}=\frac{1}{{\lambda }_3+\Delta {\lambda }_3}$$

i.e.

(38)$$\Delta {\lambda }_3=\frac{({\lambda }_1+\Delta {\lambda }_1)({\lambda }_2+\Delta {\lambda }_2)}{{\lambda }_1+\Delta {\lambda }_1+{\lambda }_2+\Delta {\lambda }_2}-{\lambda }_3$$

For SHG, a special case of three wave frequency mixing, there is λ₁ = λ₂, so λ₃ = λ₁/2 from the Eq. (36), and Δλ₃ = Δλ₁/2 from the Eq. (38).

Around the central PM wavelength λ₁₀, the mismatch of wave vector Δk can be expressed as the followed Taylor series about the wavelength λ₁

(39)$$\Delta k=\Delta k|{}_{{\lambda }_1={\lambda }_{10}}+\frac{\partial \Delta k}{\partial {\lambda }_1}|{}_{{\lambda }_1={\lambda }_{10}}\Delta {\lambda }_1+\frac{1}{2}\frac{{\partial }^2\Delta k}{\partial {\lambda }_1{}^2}|{}_{{\lambda }_1={\lambda }_{10}}(\Delta {\lambda }_{1)}{}^2+\cdot \cdot \cdot$$

At the central PM wavelength λ₁₀, there is $\Delta k|{}_{{\lambda }_1={\lambda }_{10}}=0$. To the angular NCPM concerned in this paper, the parameter $\partial \Delta k/\partial {\lambda }_1$ is usually not equal to zero, so we can solely reserve this term and ignore the higher order derivative terms, for the convenience of the calculation. Thus,

(40)$$\Delta k=\frac{\partial \Delta k}{\partial {\lambda }_1}|{}_{{\lambda }_1={\lambda }_{10}}\Delta {\lambda }_1$$

Based on the expression of wave vector, i.e. k = 2πn/λ, there is

(41)$$\Delta k=K_3-K_2-K_1=\frac{2\pi }{{\lambda }_3}{n}_3-\frac{2\pi }{{\lambda }_2}{n}_2-\frac{2\pi }{{\lambda }_1}{n}_1$$

So

(42)$$\frac{\partial \Delta k}{\partial {\lambda }_1}=2\pi \left\{\left[\frac{\frac{\partial n_3}{\partial {\lambda }_1}}{{\lambda }_3}+n_3(-\frac{1}{{\lambda }_3^2}\frac{\partial {\lambda }_3}{\partial {\lambda }_1})\right]-\left[\frac{\frac{\partial n_2}{\partial {\lambda }_1}}{{\lambda }_2}+n_2(-\frac{1}{{\lambda }_2^2}\frac{\partial {\lambda }_2}{\partial {\lambda }_1})\right]-\left[\frac{\frac{\partial n_1}{\partial {\lambda }_1}}{{\lambda }_1}+n_1(-\frac{1}{{\lambda }_1^2}\frac{\partial {\lambda }_1}{\partial {\lambda }_1})\right]\right\}$$

Considering λ₁ = λ₂ and λ₃ = λ₁/2, there are $\partial {\lambda }_3/\partial {\lambda }_1=1/2$, $\partial {\lambda }_2/\partial {\lambda }_1=1$, and $\partial {\lambda }_1/\partial {\lambda }_1=1$. So the Eq. (42) can be simplified as

(43)$$\frac{\partial \Delta k}{\partial {\lambda }_1}=2\pi \left(\frac{2\frac{\partial n_3}{\partial {\lambda }_1}-\frac{\partial n_2}{\partial {\lambda }_1}-\frac{\partial n_1}{\partial {\lambda }_1}}{{\lambda }_1}-\frac{2n_3-n_2-n_1}{{\lambda }_1^2}\right)$$

At the PM condition λ = λ₁₀, there is 2n₃ = n₂ + n₁, so

(44)$$\frac{\partial \Delta k}{\partial {\lambda }_1}=\frac{2\pi }{{\lambda }_1}\left(2\frac{\partial n_3}{\partial {\lambda }_1}-\frac{\partial n_2}{\partial {\lambda }_1}-\frac{\partial n_1}{\partial {\lambda }_1}\right)$$

Because we only consider the angular NCPM styles in this paper, all of n₃, n₂, and n₁ are principal refractive indexes. The principal refractive indexes of biaxial crystal are

(45)$$n_j^2({\omega }_i)=A_j+\frac{B_j}{{\lambda }_i^2-C_j}-D_j{\lambda }_i^{2}j=X,Y,Zi=1,2,3$$

So $\partial n_j({\omega }_i)/\partial {\lambda }_1$ can be derived as

(46)$$\frac{\partial n_j({\omega }_i)}{\partial {\lambda }_1}=\frac{-\frac{B_j}{{({\lambda }_i^2-C_j)}^2}-D_j}{\sqrt{A_j+\frac{B_j}{{\lambda }_i^2-C_j}-D_j{\lambda }_i^2}}{\lambda }_i\frac{\partial {\lambda }_i}{\partial {\lambda }_1}$$

where $\partial {\lambda }_3/\partial {\lambda }_1=1/2$, $\partial {\lambda }_2/\partial {\lambda }_1=1$, and $\partial {\lambda }_1/\partial {\lambda }_1=1$.

Considering the equation (40) and the definition of the wavelength acceptance bandwidth, we can obtain

(47)$$\Delta k=\frac{\partial \Delta k}{\partial {\lambda }_1}|{}_{{\lambda }_1={\lambda }_{10}}\Delta {\lambda }_1=\pm \frac{\pi }{l}$$

i.e.

(48)$$\Delta {\lambda }_{1}l=2\pi \left|{(\frac{\partial \Delta k}{\partial {\lambda }_1}|{}_{{\lambda }_1={\lambda }_{10}})}^{-1}\right|$$

where $\frac{\partial \Delta k}{\partial {\lambda }_1}|{}_{{\lambda }_1={\lambda }_{10}}$ can be obtained from the Eqs. (44) and (46).

Below we will discuss the type-I NCPM SHG on Y-axis of biaxial crystal as an example. From the Eq. (46), there is

(49)$$\frac{\partial n_Z({\omega }_1)}{\partial {\lambda }_1}=\frac{\partial n_Z({\omega }_2)}{\partial {\lambda }_1}=\frac{-\frac{B_Z}{{({\lambda }_1^2-C_Z)}^2}-D_Z}{\sqrt{A_Z+\frac{B_Z}{{\lambda }_1^2-C_Z}-D_Z{\lambda }_1^2}}{\lambda }_1$$

(50)$$\frac{\partial n_X({\omega }_3)}{\partial {\lambda }_1}=\frac{-\frac{B_X}{{({\lambda }_3^2-C_X)}^2}-D_X}{\sqrt{A_X+\frac{B_X}{{\lambda }_3^2-C_X}-D_X{\lambda }_3^2}}(\frac{{\lambda }_3}{2})$$

For type-I NCPM SHG, n₃ = n_X (ω₃), n₂ = n₁ = n_Z(ω₁) in the Eq. (44). Taking Eq. (49) and (50) into the Eq. (44), we can achieve

$$\frac{\partial \Delta k}{\partial {\lambda }_1}=\frac{2\pi }{{\lambda }_1}\left(\frac{-\frac{B_X}{{({\lambda }_3^2-C_X)}^2}-D_X}{\sqrt{A_X+\frac{B_X}{{\lambda }_3^2-C_X}-D_X{\lambda }_3^2}}{\lambda }_3-2\frac{-\frac{B_Z}{{({\lambda }_1^2-C_Z)}^2}-D_Z}{\sqrt{A_Z+\frac{B_Z}{{\lambda }_1^2-C_Z}-D_Z{\lambda }_1^2}}{\lambda }_1\right)$$

$$=\frac{2\pi }{{\lambda }_1}\left(\frac{-\frac{B_X}{{({\lambda }_3^2-C_X)}^2}-D_X}{n_X({\omega }_3)}{\lambda }_3-2\frac{-\frac{B_Z}{{({\lambda }_1^2-C_Z)}^2}-D_Z}{n_Z({\omega }_1)}{\lambda }_1\right)$$

(51)$$=\pi [4\frac{\frac{B_Z}{{({\lambda }_1^2-C_Z)}^2}+D_Z}{n_Z({\omega }_1)}-\frac{\frac{B_X}{{({\lambda }_3^2-C_X)}^2}+D_X}{n_X({\omega }_3)}]$$

Taking Eq. (51) into the Eq. (48), the wavelength acceptance bandwidth of type-I NCPM SHG is obtained to be

$$\Delta {\lambda }_{1}l=2\left|{[4\frac{\frac{B_Z}{{({\lambda }_1^2-C_Z)}^2}+D_Z}{n_Z({\omega }_1)}-\frac{\frac{B_X}{{({\lambda }_3^2-C_X)}^2}+D_X}{n_X({\omega }_3)}]}^{-1}|{}_{{\lambda }_1={\lambda }_{10}}\right|$$

$$=2\left|{[4\frac{\frac{B_Z}{{({\lambda }_{10}^2-C_Z)}^2}+D_Z}{n_Z({\omega }_{10})}-\frac{\frac{16B_X}{{({\lambda }_{10}^2-4C_X)}^2}+D_X}{n_X({\omega }_{30})}]}^{-1}\right|$$

(52)$$=2\left|{[\frac{\frac{4B_Z}{{({\lambda }_{10}^2-C_Z)}^2}+4D_Z-\frac{16B_X}{{({\lambda }_{10}^2-4C_X)}^2}-D_X}{n_Z({\lambda }_{10})}]}^{-1}\right|$$

Similarly, the wavelength acceptance bandwidth for other five kinds of NCPM SHG configurations can also be deduced out, which are listed in the Table 9.

Table 9. Wavelength acceptance bandwidth of NCPM SHG in biaxial NLO crystal

View Table | View all tables in this article

12.4 Temperature acceptance bandwidth of NCPM SHG

The variations of the principal refractive indexes with the temperature can be expressed by

(53)$$n_X(\lambda )=n_{0X}(\lambda )+\frac{\partial n_X(\lambda )}{\partial T}\Delta T$$

(54)$$n_Y(\lambda )=n_{0Y}(\lambda )+\frac{\partial n_Y(\lambda )}{\partial T}\Delta T$$

(55)$$n_Z(\lambda )=n_{0Z}(\lambda )+\frac{\partial n_Z(\lambda )}{\partial T}\Delta T$$

where n_0X(λ), n_0Y(λ), and n_0Z(λ) are the principal refractive indexes at normal atmospheric temperature T₀. The wave vector mismatching for three wave frequency mixing is

(56)$$\Delta k=k_3-k_2-k_1=\frac{2\pi }{{\lambda }_3}{n}_3-\frac{2\pi }{{\lambda }_2}{n}_2-\frac{2\pi }{{\lambda }_1}{n}_1$$

So, its first order derivative with respect to temperature is given by

(57)$$\frac{\partial \Delta k}{\partial T}=\frac{2\pi }{{\lambda }_3}\frac{\partial n_3}{\partial T}-\frac{2\pi }{{\lambda }_2}\frac{\partial n_2}{\partial T}-\frac{2\pi }{{\lambda }_1}\frac{\partial n_{{}_1}}{\partial T}$$

For SHG where λ₁ = λ₂, λ₃ = λ₁/2, there is

(58)$$\frac{\partial \Delta k}{\partial T}=\frac{2\pi }{{\lambda }_1}(2\frac{\partial n_3}{\partial T}-\frac{\partial n_2}{\partial T}-\frac{\partial n_{{}_1}}{\partial T})$$

The mismatch of wave vector Δk can be expressed as the followed expression about the temperature

(59)$$\Delta k=\frac{\partial \Delta k}{\partial T}|{}_{T=T_0}\Delta T$$

Considering the definition of the temperature acceptance bandwidth (Δk = ± π/l), we can obtain

$$\Delta Tl=2\pi \left|{(\frac{\partial \Delta k}{\partial T}|{}_{T=T_0})}^{-1}\right|$$

(60)$$=\frac{{\lambda }_1}{2\frac{\partial n_3}{\partial T}-\frac{\partial n_2}{\partial T}-\frac{\partial n_{{}_1}}{\partial T}}$$

For the angular NCPM SHG considered in this paper, all of n₃, n₂, and n₁ are principal refractive indexes. So $\frac{\partial n_3}{\partial T}$, $\frac{\partial n_2}{\partial T}$, and $\frac{\partial n_1}{\partial T}$ are the principal refractive index−temperature coefficients, as demonstrated in the Eqs. (53)-(55).

For the type-I angular NCPM SHG on Y-axis of biaxial crystal, there are n₃ = n_X(λ₃), n₁ = n₂ = n_Z(λ₁). From the Eq. (60), the temperature acceptance bandwidth at this situation is obtained to be

(61)$$\Delta Tl=\frac{{\lambda }_1}{2\left[\frac{\partial n_X({\lambda }_3)}{\partial T}-\frac{\partial n_Z({\lambda }_1)}{\partial T}\right]}$$

Similarly, the temperature acceptance bandwidth for other five kinds of angular NCPM SHG configurations can also be deduced out, which are listed in the Table 10.

Table 10. Temperature acceptance bandwidth of NCPM SHG in biaxial NLO crystals

View Table | View all tables in this article

13. Funding

National Natural Science Foundation of China (NSFC) (Grant Nos. 61178060, 51202129 and 91022034); Nature Science Foundation of Shandong Province (ZR2012EMQ004); Natural Science Foundation for Distinguished Young Scholar of Shandong Province (2012JQ18).

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